Robust Fuzzy Output Feedback Controller for Affine Nonlinear Systems via T–S Fuzzy Bilinear Model - CSTR Benchmark

Robust Fuzzy Output Feedback Controller for

Affine Nonlinear Systems via T–S Fuzzy Bilinear

Model: CSTR Benchmark

By M. Hamdy, I. Hamdan

Presentation by Mostafa Shokrian Zeini

Important Questions:

- What is a T-S fuzzy bilinear model? And why do we have to use bilinear model of systems?

- How to design a robust fuzzy controller based on PDC for stabilizing the T-S fuzzy bilinear model with disturbance?

- Why do we use an output feedback controller instead of a state feedback one?

- What are the conditions on the control parameters and how to derive them?

2

The T-S fuzzy model is a popular adopted fuzzy model.

- it has good capability to describe a nonlinear system.- it can accurately approximate the given nonlinear systems with fewer rules than other types of fuzzy models.

Because

3

T-S Fuzzy Bilinear Model

However

• Most of the existing results focus on the stability analysis and synthesis based on T-S fuzzy model with linear local model.

when a nonlinear system cannot be adequately approximated by linear model, but bilinear model, we have to use another modelling approach.

4


A bilinear system is expressed as follows:

bilinear systems involve products of state and control.

means that they are linear in state and linear in control, but not jointly linear in state and control .

Obviously

5


Which

Bilinear systems naturally represent many physical and biological processes.

a bilinear model can obviously represent the dynamics of a nonlinear system more accurately than a linear one.

Also

6


Robust stabilization for continuous-time fuzzy bilinear system with disturbance

Robust stabilization for continuous-time fuzzy bilinear system with time-delay only in the state

Robust fuzzy control for a class of uncertain discrete fuzzy bilinear system

Robust fuzzy control for a class of uncertain discrete fuzzy bilinear system with time-delay only in the state

7

T-S Fuzzy Bilinear Model and Fuzzy Controller Design

8

All the results were obtained based on either state feedback controller

or observer-based controller.

T-S Fuzzy Bilinear Model and Fuzzy Controller Design

in many practical control problems, the physical state variable of systems is partially or fully unavailable for

measurement

Since the state variable is not accessible for sensing devices

and transducers are not available or very expensive:

9

Output Feedback Controller

10

In such cases, the scheme of output feedback controller is very

important and must be used when the system states are not

completely available for feedback.

Output Feedback Controller


The T-S fuzzy bilinear model has been constructed for approximating the behaviour nonlinear systems with disturbance

in the neighborhood of the desired equilibrium or desired operating point .

Consider a class of nonlinear system affine in the input variables:

�̇� (𝑡 )= 𝑓 (𝑥 (𝑡 ) ,𝑢 (𝑡 ) )=𝐹 (𝑥 (𝑡 ) )+𝐺 (𝑥 (𝑡 ) )𝑢 (𝑡 )+𝑁𝑥 (𝑡 )𝑢 (𝑡 )+𝐸𝑤 (𝑡)

11

The following condition should be satisfied:

𝐹 (𝑥 (𝑡 ) )+𝐺 (𝑥 (𝑡 ) )𝑢 (𝑡 )≅ 𝐴𝑥 (𝑡 )+𝐵𝑢(𝑡)

𝑥(𝑡)=𝐴𝑥(𝑡)+𝐵𝑢(𝑡)+𝑁𝑥(𝑡)𝑢(𝑡)+𝐸𝑤(𝑡)From above, the values of the matrices and are used as the same values.

12

T-S Fuzzy Bilinear Model�̇� (𝒕 )= 𝒇 (𝒙 (𝒕 ) ,𝒖 (𝒕 ) )=𝑭 (𝒙 (𝒕 ) )+𝑮 (𝒙 (𝒕 ) )𝒖 (𝒕 )+𝑵𝒙 (𝒕 )𝒖 (𝒕 )+𝑬𝒘 (𝒕 )

Let be the th row of the matrix and be the th component of :

𝑓 𝑖 (𝑥 (𝑡 ) )≅𝑎𝑖𝑇 𝑥 (𝑡 ) , 𝑖=1 ,2 ,…,𝑛

The matrix has been deduced as the same value from affine nonlinear system.The matrix has been changed in each desired equilibrium point.

Thus, constant matrices and should be find such that in a neighborhood of : 𝐹 (𝑥 )=𝐴𝑥 ,𝐹 (𝑥𝑑 )=𝐴 𝑥𝑑 ;𝐺 (𝑥 )𝑢 (𝑡 )=𝐵𝑢 (𝑡 ) ,𝐺 (𝑥𝑑)=𝐵

13


�̇� (𝒕 )=𝑨𝒙 (𝒕 )+𝑩𝒖 (𝒕 )+𝑵𝒙 (𝒕 )𝒖 (𝒕 )+𝑬𝒘 (𝒕)

From the above equations:

𝛻 𝑥𝑇 𝑓 𝑖 (𝑥𝑑 )≅ 𝑎𝑖

𝑇

Expanding the left-hand side of and neglecting h.o.t.: 𝑓 𝑖 (𝑥𝑑)+∇𝑥

𝑇 𝑓 𝑖 (𝑥𝑑) (𝑥−𝑥𝑑)≅ 𝑎𝑖𝑇 𝑥 (𝑡 )

At the operating point:

𝑓 𝑖 (𝑥𝑑)≅ 𝑎𝑖𝑇𝑥𝑑 ,𝑖=1 ,2 , …,𝑛

14



One can reformulate the objective as a convex constrained optimization problem:

One should determine such that is “close to” in the neighborhood of . Consider the following performance index:

15



Substituting into :

Pre-multiplying by and substituting into the resulting equation yield:

16



17

This completes the construction of matrices , , , and in each desired

equilibrium point for T-S fuzzy bilinear model from affine nonlinear system

with disturbance.



Let’s derive the T-S fuzzy bilinear model: Plant rule i:

IF is , and … … and is , THEN

18


By using singleton fuzzifier, product inference and center-

average defuzzifier, then:the T-S fuzzy bilinear model is described by the following

global model:

19


20


Control rule :IF is , and … … and is THEN

The fuzzy controller is designed to stabilize the T-S fuzzy bilinear model with disturbances.

The th rule of the robust fuzzy output feedback controller:

21

Robust Fuzzy Controller Design Based on PDC

The overall T-S fuzzy controller can be formulated as follows:

22


, is a scalar to be determined by LMI conditions and is an arbitrary designed scalar, .

where

23


Rearranging the previous equation:

The closed-loop fuzzy system:

24


We introduce the following performance criterion with its control objectives:

When , the resulting of the closed-loop system is asymptotically stable.

For , and , the controlled output of the closed-loop system satisfies for all non-zero .

i

25

Robust Fuzzy Output Feedback Controller

ii

26

The Overall Block Diagram

27

The Stability Analysis and LMI

Conditions

The proposed control law should be designed such that to guarantee the asymptotic stability of the closed-loop system.

The following stability analysis is carried out to determine the LMI conditions on control parameters.

So

The time derivative of becomes:

�̇� (𝑥 (𝑡 ) )=�̇� (𝑡 )𝑇 𝑃𝑥 (𝑡 )+𝑥 (𝑡 )𝑇 𝑃 �̇�(𝑡)

We consider the following Lyapunov function candidate:

𝑣 (𝑥 (𝑡 ) )=𝑥 (𝑡 )𝑇 𝑃𝑥 (𝑡)

28

The Stability Analysis and LMI Conditions

By substituting the closed-loop system into the previous equation:

29


The performance level implies that:

30


Rearranging:

31


Lemma 1

For any two matrices and with appropriate dimensions, and , we have:

32


Using lemma 1:

33


Rearranging:

34


Hence if , then for all .

where

35


The previous matrix inequality is quadratic matrix inequality

(QMI).The Schur complement is used to transform the QMI to LMI:

36


The previous matrix inequality is bilinear matrix inequality (BMI), because of the product of two terms and which is

bilinear.To transform to LMI, we define a new variable :

37


Now, we can summarize the overall design procedure of the proposed scheme in the following three steps:

Let the parameters , , and in the derived LMI condition.

Solve the derived LMI to obtain positive definite matrix , and the controller gains .

Apply the robust fuzzy control law into the T-S fuzzy bilinear model; one can get the closed-loop fuzzy system.

2

38

Robust Fuzzy Output Feedback Controller

3

1

The Continuous Stirred Tank Reactor (CSTR) benchmark has widespread application in industry and is often characterized

by highly nonlinear behavior.Consider the nonlinear model for dynamics of an isothermal CSTR benchmark with disturbance given by:

39

Simulation Results

40

Simulation Results

41

Simulation Results

Based on the proposed T-S fuzzy bilinear modeling, all the system matrices are

constructed as follows:

42

Simulation Results

Based on the proposed T-S fuzzy bilinear modeling, all the system matrices are

constructed as follows:

43

Simulation Results

The proposed scheme design procedure is described in the following steps:

Let the parameters , , and in the LMI.

Solve the derived LMI, we obtain positive definite matrix , and the controller gains , , .

Using all the data from the previous steps, we can construct the fuzzy control law, and the initial condition is chosen as .

2

44

3

1

Simulation Results

45

Simulation Results

46

Simulation Results

47

Simulation Results

48

Simulation Results

49

Simulation Results

50

Simulation Results

51

Simulation Results

52

Simulation Results

53

References

1. M. Hamdy, I. Hamdan, “Robust Fuzzy Output Feedback Controller for Affine Nonlinear Systems via T–S Fuzzy Bilinear Model: CSTR Benchmarkˮ, 2015, ISA Transactions, In Press.

2. K. Tanaka, H. O. Wang, “Fuzzy Control Systems Design and Analysis - A Linear Matrix Inequality Approachˮ, John Wiley & Sons, New York, 2001.

3. T.H.S. Li, S.H. Tsai, “T-S Fuzzy Bilinear Model and Fuzzy Controller Design for a Class Nonlinear Systemsˮ, 2007, IEEE Transactions on Fuzzy Systems, 15(3), pp. 494-506.

4. M. Hamdy, I. Hamdan, “A New Calculation Method of Feedback Controller Gain for Bilinear Paper-Making Process with Disturbanceˮ, 2014, J. Process Control, 24, pp. 1402-1411.

Engineering

Robust Fuzzy Output Feedback Controller for Affine Nonlinear Systems via T–S Fuzzy Bilinear Model - CSTR Benchmark